Homotopy and crossings of systems of curves on a surface

AbstractLet C1,…,Ck and C′1,…,C′k be closed curves on a compact orientable surface S. We characterize (in terms of counting crossings) when there exists a permutation π of {1,…,k} such that, for each i=1,…,k,C′π(i) is freely homotopic to Ci or C-1i characterization is equivalent to the nonsingularity of a certain infinite symmetric matrix

On the uniqueness of kernels

AbstractLet S be a compact orientable surface. For any graph G embedded on S and any closed curve D on S we define μG(D) as the minimum number of intersections of G and D′, where D′ ranges over all closed curves freely homotopic to D. We call G a kernel if μG′ ≠ μG for each proper minor G′ of G. We prove that if G and G′ are kernels with μG = μG′ (in such a way that each face of G is an open disk), then G′ can be obtained from G by a series of the following operations: (i) homotopic shifts over S; (ii) taking the surface dual graph; (iii) ΔY-exchange (i.e., replacing a vertex v of degree 3 by a triangle connecting the three vertices adjacent to v, or conversely)

Circuits in graphs embedded on the torus

AbstractWe give a survey of some recent results on circuits in graphs embedded on the torus. Especially we focus on methods relating graphs embedded on the torus to integer polygons in the Euclidean plane